The best account (as so often happens) is in Wikipedia, which says it "recall[s] the escape of Prince Charles Edward Stuart (Bonnie Prince Charlie) from Uist to the Isle of Skye after his defeat at the Battle of Culloden in 1746." Don't believe the bad things you hear about Wikipedia!
According to Wikipedia, for all their romance about Scottish rebellion, the lyrics was written by the British aristocracy of the 1800s -- " by Sir Harold Boulton, 2nd Baronet, to an air collected in the 1870s by Anne Campbelle MacLeod (1855–1921), who became Lady Wilson by marriage to Sir James Wilson KSCI (1853–1926) in 1888." The air, or melody, is traditional, however, and it's lovely. It has been covered by everyone from James Galway and the Chieftans to Tom Jones and Rod Stewart. YouTube has covers by the Corries, the Scots Guards Pipe and Drums and the Tampa Bay Children's Chorus, among others.
A lead sheet is available on EverythingDulcimer.com at http://www.everythingdulcimer.com/tab/ (scroll down the alphabetical directory to the "rlwalker" version of "Skye Boat Song.") It's dulcimer tab, in DAD and DAA, but it has the melody line and chords for those of us who play other instruments. You can also print it out directly by linking here:
There are several versions on YouTube, but the one that's probably the most instructive for us shows a local band called Carl Purdy and Friends at O'Donovan's Irish Pub in Augusta, Ga., Musicians include: Carl (harmony, Irish pennywhistle) and Erin (lead vocal, guitar). The woman who inquires at 1:50 who ordered the "burger with a side of mayonnaise" is surely a waitress and not part of the performance. Think of her as adding to the ambiance!
There's also a nice fingerpicked version on the mountain dulcimer by YouTube user dulcibard.
A classical arrangement -- Beautiful Classic Scottish Music - The Skye Boat Song - Relaxing Harp, Flute, Clarinet, Violin Solo: Beautiful instrumental Scottish folk music solo of the classic traditional Skye Boat - Speed Bonnie Boat song. Best Relaxing Harp and Peaceful Flute music. Mormon Tabernacle Choir Principal Flute, Jeannine Goeckeritz and Principal Harp, Tamara Oswald performing live an arrangement by Skaila Kanga and Clive Romney, joined by Becca Goeckeritz, violin, Daron Bradford, clarinet, and Scott Allen, bass. From the Oswald Goeckeritz Duo live concert "A Joyful Evening" at BYU Education Week. http://www.harpandflute.com.
And an a cappella performance by the UCD Choral Scholars at University College Dublin. If you'd like to hear the lyrics without a conversation about hamburgers in the background, this is the one to listen to!